Amazon's app received an update today, bringing Prime Instant Video streaming to Android devices. To enable streaming, you have to download the Prime Instant Video player, which is available exclusively in Amazon's appstore.
Here's the changelog detailing all the new features:

US customers can now use a single app to discover and shop for hundreds of millions of physical and digital items.

Prime members in the US can stream unlimited Prime Instant Video from inside the Amazon app using the Prime Instant Video player, including HBO shows The Sopranos, Boardwalk Empire, Six Feet Under, and The Wire

Customers can shop any of Amazon's sites around the world from a single app

Quickly search, get product details, and read reviews on millions of products sold by Amazon and other merchants